Constraints-based models: Regulation of Gene Expression Reduces the Steady-state Solution Space

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1 J. theor. Biol. (2003) 221, doi: /jtbi , available online at on Constraints-based models: Regulation of Gene Expression Reduces the Steady-state Solution Space Markus W. Covertw and Bernhard O. Palsson n w wdepartment of Bioengineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA , U.S.A. (Received on 17 December 2001, Accepted in revised form on 25 April 2002) Constraints-based models have been effectively used to analyse, interpret, and predict the function of reconstructed genome-scale metabolic models. The first generation of these models used hard non-adjustable constraints associated with network connectivity, irreversibility of metabolic reactions, and maximal flux capacities. These constraints restrict the allowable behaviors of a network to a convex mathematical solution space whose edges are extreme pathways that can be used to characterize the optimal performance of a network under a stated performance criterion. The development of a second generation of constraintsbased models by incorporating constraints associated with regulation of gene expression was described in a companion paper published in this journal, using flux-balance analysis to generate time courses of growth and by-product secretion using a skeleton representation of core metabolism. The imposition of these additional restrictions prevents the use of a subset of the extreme pathways that are derived from the hard constraints, thus reducing the solution space and restricting allowable network functions. Here, we examine the reduction of the solution space due to regulatory constraints using extreme pathway analysis. The imposition of environmental conditions and regulatory mechanisms sharply reduces the number of active extreme pathways. This approach is demonstrated for the skeleton system mentioned above, which has 80 extreme pathways. As regulatory constraints are applied to the system, the number of feasible extreme pathways is reduced to between 26 and 2 extreme pathways, a reduction of between 67.5 and 97.5%. The method developed here provides a way to interpret how regulatory mechanisms are used to constrain network functions and produce a small range of physiologically meaningful behaviors from all allowable network functions. r 2003 Elsevier Science Ltd. All rights reserved. Introduction The recent torrent of genome annotations and subsequent metabolic reconstructions has led to the development of mathematical modeling approaches to analyse the integrated behavior of microbial cells on a genome scale (Tomita et al., n Corresponding author. Tel.: ; fax: address: palsson@ucsd.edu (B.O. Palsson). 1999; Gombert & Nielsen, 2000; Varner, 2000; Covert et al., 2001a). A constraints-based approach to studying metabolic models in silico has proven effective in the analysis of these genome-scale models (Palsson, 2000). The statement of governing constraints defines a solution space within which optimal solutions can be found using linear optimization (Varma & Palsson, 1994a; Bonarius et al., 1997; Edwards et al., 1999). This constraints-based approach /03/$35.00 r 2003 Elsevier Science Ltd. All rights reserved.

2 310 M. W. COVERT AND B. O. PALSSON has been useful in generating hypotheses in silico which may be tested experimentally (Varma & Palsson, 1994b, c). The result thus far has been a surprising degree of correlation between the predictions of genome-scale models and independently obtained experimental data (Edwards & Palsson, 2000; Edwards et al., 2001). The study of the characteristics of the solution space uses principles of convex analysis (Schilling et al., 1999; Schilling & Palsson, 2000). Extreme pathways are calculated as edges of the solution space, where the optimal solution must lie (Schilling et al., 2000a). Taken together, the formulation of hard constraints associated with network connectivity, reaction irreversibility, and maximal flux restrictions, has led to the formulation and testing of what might be called the first generation of constraints-based models of biochemical reaction networks. A logical next step in adding to the functionality of these genome-scale models is to incorporate regulation of gene expression. Several approaches have been used to model metabolic and regulatory events, beginning with Boolean representation of genetic circuits (Thomas, 1973, 1991; Kaufman et al., 1985; Kauffman, 1993; Thieffry & Thomas, 1995; Somogyi & Sniegoski, 1996), which is advantageously simple but lacks the ability to make quantitative predictions. On the other hand, deterministic approaches (Reich & Sel kov, 1981; Fell, 1996; Heinrich & Schuster, 1996; Wong et al., 1997; Stephanopoulos et al., 1998) may be combined with Boolean logic (McAdams & Shapiro, 1995), fuzzy logic (Lee et al., 1999), or cybernetic principles (Kompala et al., 1986; Varner, 2000) to model regulatory and metabolicevents. Stochasticmodeling approaches have also been developed to account for the low concentrations, spatial isolation, and slow reaction rates which characterize metabolic reactions in a single cell (McAdams & Arkin, 1997, 1998, 1999). Both deterministicand stochastic modeling approaches have been limited by the lack of experimental methods to determine kineticand other parameters, as well as the computational difficulties of developing these models on a genome-scale (Bailey, 2001). The constraints-based approach to modeling microbial metabolism has recently been expanded to incorporate regulatory constraints (Covert et al., 2001b). Regulatory constraints differ from the rigid physico-chemical constraints in two important ways; they are (1) self-imposed, meaning that over time, evolution has selected the development of complex mechanisms to restrict the allowable behaviors of these organisms under various conditions; and (2) time dependent, in that the state of the external and internal environment at a given time point determines transcriptional activity. As a result of these two features, the effects of transcriptional regulation can be treated as temporary constraints on the metabolic system. These constraints reduce the size of the solution space and change its shape from one environmental condition to another (Fig. 1). In this paper we continue the work described in an earlier manuscript published in this journal (Covert et al., 2001b), where regulatory events and gene expression were described using Boolean logicequations. For a given environment, we determine the corresponding regulatory constraints (e.g. repression of gene transcription) and eliminate extreme pathways that are inconsistent with the imposed regulatory constraints. This procedure reduces the solution space and customizes it for the given environmental conditions. The inclusion of known regulatory mechanisms effectively moves us toward the formulation of second-generation constraintsbased models of complex biochemical reaction networks: models that combine metabolic flux-balance formalism and regulation of gene expression. Methods SAMPLE METABOLIC NETWORK A skeleton network of core metabolism was formulated earlier (Covert et al., 2001b). It includes 20 reactions, seven of which are governed by regulatory logic. This network is a highly simplified representation of core metabolic processes (e.g. glycolysis, the pentose phosphate pathway, TCA cycle, fermentation pathways, amino acid biosynthesis and cell growth), along with corresponding regulation (e.g. catabolite repression, aerobic/anaerobic regulation, amino acid biosynthesis regulation and carbon storage regulation). A schematic of

3 REGULATION: SOLUTION SPACE REDUCTION 311 Flux A Flux A Flux C Flux C P 2 P 1 P 2 this skeleton network is shown in Fig. 2, together with a table containing all of the relevant chemical reactions and regulatory rules which govern the transcriptional regulation. EXTREME PATHWAYS The steady-state behavior of a metabolic system may be characterized as Sv ¼ 0; ð1þ where S mn (m metabolites n fluxes) is the stoichiometric matrix for the network and v n1 is a vector of the flux levels through each reaction in the system. Certain constraints on the system, P 3 P 4 Flux B P 3 P 4 P1 P2 P3 P4 P1 is not permitted due to regulatory constraints Flux B Fig. 1. Regulatory constraints reduce the steady-state solution space of a metabolic network. A solution space bounded by invariant constraints on the network is shown. Extreme pathways may be calculated as the unique, systemically independent generating vectors for the space. In the space on top, all of the pathways are considered available to the system (denoted by the highlighted gray boxes at right). Under certain environments however, regulatory constraints may cause one or more of the extreme pathways to be temporarily unavailable to the system, P1 in the case shown here. This results in a more restricted space with a reduced volume and/or dimension (bottom), corresponding to a metabolic network with fewer available behaviors. P1 P2 P3 P4 such as the thermodynamics of reactions or the constraints associated with enzyme capacity (e.g. maximum metabolite uptake and/or secretion rates) may be represented as upper and lower bounds on reaction flux levels. If reaction fluxes are also constrained to positive values by decomposing reversible reactions in S into forward and reverse components, a solution space may be geometrically defined for the system as a convex polyhedral cone in n-dimensional space (Schilling et al., 2000a). Such a space contains every possible steady-state flux distribution available to the system, subject to the given constraints. A minimal set of generating vectors may be determined for the convex polyhedral cone using principles of convex analysis. This set is approximately analogous to a basis in linear algebra. The generating vectors span the null space of S and are the edges of the cone (Schilling & Palsson, 1998). Given that the cone represents the metabolicflux solution space at steady state, each generating vector or extreme ray corresponds to a particular pathway or active set of fluxes in the metabolicnetwork and is termed an extreme pathway. Every possible steady-state flux distribution of a metabolicnetwork may therefore be represented as a positive combination of extreme pathways: ( ) C ¼ v : v ¼ Xk a i p i ; a i X0; 8i ; ð2þ i¼1 where C is the polyhedral cone representing the metabolicnetwork at steady state and p i represents the extreme pathway vectors (Schilling et al., 1999). The algorithm used to generate extreme pathways has been described in detail (Schilling et al., 2000b). BOOLEAN LOGIC DESCRIPTION OF ENVIRONMENTAL AND REGULATORY CONSTRAINTS Boolean logicformalism can be used to represent regulatory constraints (Covert et al., 2001b) where a gene is considered in one of two states: active (ON) or inactive (OFF). Here, this formalism is also used to describe environmental constraints where a substrate is considered either present in (ON) or absent from (OFF) the external medium. The regulatory rules are listed

4 312 M. W. COVERT AND B. O. PALSSON H ext 2 NADH 0.2 C R 5a,b 1 ATP T 2 NADH h H G R 8a,b Carbon 2 T c2 Carbon 1 F ext REACTION NAME REGULATION Oxygen T Metabolic Reactions c1 A + ATP B R1 Tf B + 2 NADH + C R2a IF NOT(RPb) A T C NADH B R2b O2 B F R3 ATP C G R4 R 1 O G 0.8 C + 2 NADH R5a IF NOT (RPo2) 2 G 0.8 C + 2 NADH R5b IF RPo2 R B 3 C + 3 D R6 NADH ATP C + 4 NADH 3 E R7 IF NOT (RPb) F R res G + ATP + 2 NADH H R8a IF NOT (RPh) 3 NADH R H G+ ATP + 2NADH R8b 2a NADH + O2 ATP Rres IF NOT (RPo2) Transport Processes R 2b 3 NADH Carbon1 A Tc1 Carbon2 A Tc2 IF NOT(RPc1) C R 6 Fext F Tf 3 D D Dext Td T d D E Eext Te ext Hext H Th Oxygen O2 To2 R 3 E 4 R 4 NADH 7 Maintenance and Growth Processes T C + F + H + 10 ATP Biomass Growth e E ext Regulatory Proteins R z RPo2 IF NOT(Oxygen) 10 ATP RPc1 IF Carbon1 RPh IF Th RPb IFR2b Biomass Fig. 2. A schematic of the simplified core metabolic network, together with a table containing the stoichiometry of the 20 metabolic reactions, seven of which are regulated by four regulatory proteins. This network is a highly simplified representation of core metabolic processes, including a glycolytic pathway with primary substrates carbon1 (C1) and carbon2 (C2), as well as a pentose phosphate pathway and a TCA cycle, through which amino acid H enters the system. Fermentation pathways as well as amino acid biosynthesis are also represented. The regulation modeled in this study includes simplified versions of catabolite repression (e.g. preferential uptake of C1 over C2), aerobic/anaerobic regulation, amino acid (H) biosynthesis regulation and carbon storage regulation, and is also listed. The Growth reaction is indicated by a dashed line. in Fig. 2. Using these restrictions, an extreme pathway may be determined infeasible (1) if the external nutrient is absent from the external medium (e.g. pathways that have oxygen as an input would be inoperative in a simulation in an anaerobicenvironment); or (2) when expression of the gene responsible for producing a metabolicflux has been repressed (e.g. if extracellular carbon1 is present, then pathways which utilize T c2 are considered infeasible due to the repression of T c2 transcription by carbon1). Conversely, a pathway is feasible for a certain condition if it is consistent with all the applicable environmental and regulatory constraints. PHENOTYPIC PHASE PLANE ANALYSIS Phenotypicphase planes (PhPPs) are twodimensional representations of the solution space (Edwards & Palsson, 1999; Schilling et al., 2000a). Once the axes of the PhPP are set (generally uptake rates of two nutrients such as oxygen and a carbon source), flux maps which are optimal with respect to growth may be calculated for each point in the plane. The lines in a PhPP demarcate a change in the flux map; these lines are determined using the shadow prices (the dual solution in linear programming). PhPP analysis has recently been used to demonstrate the optimal growth of E. coli on succinate and acetate minimal media (Edwards et al., 2001). Results A total of 80 extreme pathways were calculated for the simplified metabolicsystem shown in Fig. 2. They are all shown graphically in the Appendix. The numbering of the extreme pathways is arbitrary and corresponds to the order in which the algorithm generates them. Given the five inputs to the metabolicnetwork and representing these inputs using Boolean logic, considering each as ON if present or OFF if absent, there are a total of 2 5 ¼ 32 possible environments which may be recognized by the cell. These environments are listed in Table 1. For each environment, the transcription of several of the enzymes in the network may be restricted due to regulation. The constraints imposed on the metabolicsystem by both the substrates available to (i.e. the external environment) and the enzymes expressed in the cell (i.e. the internal

5 REGULATION: SOLUTION SPACE REDUCTION 313 Table 1 A list of all the possible environments which can be recognized by the system shown in Fig. 1, of which six environments do not enable the cell to produce biomass Environments Repressed enzymes Pathways Pathway list C1 C2 F H O2 R 5b R 8a T c2 26 P2, P4, P5, P6, P8, P9, P10, P12, P29, P30, P31, P32, P33, P34, P35, P36, P37, P38, P45, P46, P47, P48, P49, P50, P51, P52 Detail in Fig. 4. C1 C2 F H R 5a R 8a R res T c2 10 P39, P40, P41, P42, P43, P44, P49, P50, P51, P52 C1 C2 F O2 R 5b T c2 8 P29, P30, P33, P34, P45, P46, P49, P50 C1 C2 F R 5a R res T c2 4 P41, P42, P49, P50 C1 C2 H O2 R 5b R 8a T c2 14 P2, P5, P6, P9, P10, P30, P31, P34, P35, P37, P46, P47, P50, P51 C1 C2 H R 5a R 8a R res T c2 5 P39, P42, P43, P50, P51 C1 C2 O2 R 5b T c2 4 P30, P34, P46, P50 Detail in Fig. 3 C1 C2 R 5a R res T c2 2 P42, P50 C1 F H O2 R 5b R 8a T c2 26 P2, P4, P5, P6, P8, P9, P10, P12, P29, P30, P31, P32, P33, P34, P35, P36, P37, P38, P45, P46, P47, P48, P49, P50, P51, P52 C1 F H R 5a R 8a R res T c2 10 P39, P40, P41, P42, P43, P44, P49, P50, P51, P52 C1 F O2 R 5b T c2 8 P29, P30, P33, P34, P45, P46, P49, P50 C1 F R 5a R res T c2 4 P41, P42, P49, P50 C1 H O2 R 5b R 8a T c2 14 P2, P5, P6, P9, P10, P30, P31, P34, P35, P37, P46, P47, P50, P51 C1 H R 5a R 8a R res T c2 5 P39, P42, P43, P50, P51 C1 O2 R 5b T c2 4 P30, P34, P46, P50 C1 R 5a R res T c2 2 P42, P50 C2 F H O2 R 5b R 8a 26 P3, P4, P5, P7, P8, P9, P11, P12, P57, P58, P59, P60, P61, P62, P63, P64, P65, P66, P73, P74, P75, P76, P77, P78, P79, P80 C2 F H R 5a R 8a R res 10 P67, P68, P69, P70, P71, P72, P77, P78, P79, P80 C2 F O2 R 5b 8 P57, P58, P61, P62, P73, P74, P77, P78 C2 F R 5a R res 4 P69, P70, P77, P78 C2 H O2 R 5b R 8a 14 P3, P5, P7, P9, P11, P58, P59, P62, P63, P65, P74, P75, P78, P79 C2 H R 5a R 8a R res 5 P67, P70, P71, P78, P79 C2 O2 R 5b 4 P58, P62, P74, P78 C2 R 5a R res 2 P70, P78 F H O2 R 5b R 8a 5 P4, P5, P8, P9, P12 F H R 2a R 5a R 7 R 8a R res 0 F O2 R 5b 0 F R 5a R res 0 H O2 R 2a R 5b R 7 R 8a 2 P5, P9 H R 2a R 5a R 7 R 8a R res 0 O2 R 5b 0 0 Note: For each environment, there is a set of enzymes which are repressed under the given environmental conditions. The extreme pathways which remain feasible even under the combination of environmental and regulatory constraints are listed. For a schematic of each of the pathways, see the Appendix. C1 ¼ carbon1, C2 ¼ carbon2, O2 ¼ oxygen. Two of the environments are shown in more detail in Figs 3 and 4 and are labeled correspondingly. environment) reduce the number of extreme pathways accessible to the cell at a given time. Several interesting observations may be made from Table 1: first, 21 extreme pathways (P1, P13 P28 and P53 56, enclosed by a box in the Appendix), although stoichiometrically feasible, are always impossible due to regulatory constraints. Pathways P13 28 and P53 56 are infeasible due to the fact that R res is only expressed aerobically while R 5b is only expressed

6 314 M. W. COVERT AND B. O. PALSSON anaerobically. Therefore, any pathway that includes a flux through both R res and R 5b is eliminated. Similarly, pathways P1 and P13 are eliminated because a flux through R 2b activates a regulatory protein that represses transcription of R 7. Therefore, R 2b and R 7 cannot both be expressed together. Note that P13 is infeasible in either case. Another interesting observation from Table 1 is that several environments have identical or near-identical sets of available extreme pathways. For example, the environment containing carbon1 (C1), carbon2 (C2), F, H, and oxygen (O2) has an identical extreme pathway list to that for the environment containing C1, F, H, and O2. The reason is that T c2, the transport flux for C2, is repressed in the presence of C1. Furthermore, the extreme pathway list for the environment containing C2, F, H, and O2 is similar to the pathway lists for the previously mentioned environments, different only in that the pathways which utilize the T c1 flux in the former pathway list are replaced by pathways which utilize the T c2 flux in the latter. Finally, Table 1 shows that the highest number of extreme pathways available to the cell is 26; the lowest is 2, corresponding to a reduction in the number of available extreme pathways between 67.5 and 97.5%. A relatively simple dual-substrate environment and the most complex environment were examined in more detail to more closely investigate the effect of regulation on available pathways. EXAMPLE 1: GROWTH ON C1, C2, AND O2 The metabolicnetwork was given C1, C2, and O2 as inputs and allowed biomass, D ext and E ext as outputs. These conditions reduce the number of extreme pathways available to the cell (Fig. 3). Initially, all 80 pathways are considered, and are represented schematically in Fig. 3(a). Twentyone of the extreme pathways are always restricted by regulation, as discussed earlier; the boxes representing these pathways are darkened in gray. By considering only the pathways with appropriate inputs and outputs based on the cell environment, 49 more pathways are eliminated (shaded in light gray). Of the ten remaining pathways, six are inconsistent with the given regulation (C1 catabolite repression of the C2 transport protein T c2 or regulation due to the aerobicenvironment) (shown in black) and the flux maps for the remaining four extreme pathways are shown in Fig. 3(b). The resulting solution space is projected in three dimensions (C1 uptake rate, oxygen uptake rate and growth rate), as shown in Fig. 3(c), with the four feasible extreme pathways labeled. All the volume defined by these edges is accessible to the cell. The corresponding range of growth and uptake rates can be attained by the cell under these conditions. The two-dimensional PhPP for growth on C1 and O2 is shown in Fig. 3(d). This PhPP has two feasible phases between the lines shown, which represent the four extreme pathways available to the system. For this case, the two-dimensional projection of the extreme pathways lies on the region boundaries of the PhPP; pathways 46 and 50 are both fermentative and therefore overlap in the PhPP (oxygen uptake rate ¼ 0). Pathway 30 is the line of optimality (Edwards et al., 2001, 2002) as none of the carbon is lost in secretion of by-products; pathway 34 includes secretion of D ext and therefore gives a lower biomass yield (inset) than pathway 30. Fig. 3. Extreme pathway reduction by constraints, using growth of the sample system in a C1 and C2 aerobic medium as an example. (a) The 80 total extreme pathways calculated for the system are represented by a grid, arranged to correspond with the pathway figures in the Appendix. The number of pathways is reduced by 21 when pathways are removed which are always inconsistent with the regulatory rules (dark gray), then by 49 due to the specific environmental constraints (light gray), then by 6 as the regulation corresponding to the specific environment is considered (black). The four remaining pathways which are consistent with all the regulatory and environmental constraints are shown schematically in (b), where the thick dark arrows represent active fluxes. (c) The solution space of the system, projected on a three-dimensional space, with the pathways and the line of optimality (the pathway with the greatest growth yield) noted. (d) A two-dimensional projection of the space, superimposed on a two-dimensional phenotypic phase plane for the C1 and oxygen uptake. The region at left lies outside of the space and is therefore infeasible. "

7 REGULATION: SOLUTION SPACE REDUCTION 315 (a) All possible extreme pathways Inconsistent regulation Environment-specificity: C1, C2 and O2 Hext 2 NADH 0.2 C T h H R8a Carbon 2 Carbon 1 F ext T c1 T c2 T f A R 1 ATP R B 3 F 3 NADH R 2a R5a R 2b 3 NADH G Pathway 30 Pathway 34 R 4 R z C 4 NADH 10 ATP R 6 R 7 Oxygen T O2 NADH O 2 3 D 3 E R res T e T d ATP E ext Dext 2 NADH 0.2 C T h Hext H R8a Carbon 2 T c2 R 5a,b G T c1 A ATP R 1 R B 3 F 3 NADH R 2a R 2b 3 NADH R 4 Carbon 1 R z C 4 NADH 10 ATP F ext Tf R 7 R 6 Oxygen T O2 NADH O 2 3 D 3 E Environmentspecific regulation: R5b, Tc2 Rres T e T d ATP E ext D ext Biomass Biomass 2 NADH 0.2 C Carbon 2 T c2 R5a Pathway 46 Carbon 1 Fext T c1 Tf A ATP R 1 R 3 B F 3 NADH R 2a R 2b 3 NADH C Oxygen T O2 O 2 NADH Rres 3 D R 6 T d ATP Dext 2 NADH 0.2 C Carbon 2 T c2 R 5a,b Pathway 50 Carbon 1 T c1 Tf A ATP R 1 R 3 B F 3 NADH R 2a R 2b 3 NADH C Fext Oxygen T O2 O 2 NADH Rres 3 D R 6 T d ATP D ext Hext T h H R8a G R 4 R z 4 NADH 10 ATP R 7 3 E T e Eext Hext T h H R8a G R 4 R z 4 NADH 10 ATP R 7 3 E T e E ext (b) Biomass Biomass

8 316 M. W. COVERT AND B. O. PALSSON EXAMPLE 2: GROWTH ON C1, C2, F, H, AND O2 The allowable extreme pathways for growth on a medium containing C1, C2, F, and H in an aerobicenvironment were determined. For this case, the environment offers no restrictionsfall possible inputs are availablefand therefore the restriction of the solution space by elimination of extreme pathways is entirely due to regulatory effects. From Table 1 it is shown that R 2a,R 5b, R 7,R 8a, and T c2 are constrained to zero by the regulatory rules. Consequently, the 33 corresponding extreme pathways were removed from the solution space, resulting in a list of 26 available pathways which may be used by the cell under these conditions [Fig. 4(b)]. These remaining pathways, normalized by the total input of C1, C2, F, and H for comparison purposes, are shown in Fig. 4(a). The 26 allowable pathways shown in Fig. 4(a) may be grouped by biomass yield. In the top two sets, all pathways are optimal or very nearoptimal in terms of biomass yield, with no by-product secretion. The middle pathway sets involve secretion of either D or E while biomass is generated, and the bottom set of pathways represents purely fermentative use of the network. Again, it is seen that even with a higher number of allowable extreme pathways, the actual degree of variation in possible network behavior is surprisingly small once regulatory constraints are taken into account. The multiplicity of extreme pathways with near-optimal biomass yield gives the metabolicnetwork robustness characteristics as the cell has many alternatives with nearly the same outcome. The reduced solution space is projected in three and two dimensions, as shown in Fig. 4(c) and (d), respectively. The dimensions are the same as in Fig. 3, with the exception of the C1 uptake rate axis, which has been replaced by a normalized axis of all possible routes for substrate uptake (C1+C2+F+H). The threedimensional projection is bounded by the solid black vectors, many of which overlap in the projection (note that these extreme pathways do not actually overlap; however, in a projection of the high-dimensional solution space onto a lower dimension, they share certain characteristics such as the relationship between growth, substrate uptake rate, and oxygen uptake rate). Five pathways (P4, P9, P10, P34, P35, shown with dashed black lines) lie inside the three-dimensional projection. The structure of the solution space is more complex as compared to the simple growth condition analysed in Example 1. The solution spaces are compared on the same axis to illustrate the concept of solution space reduction further [Fig. 5(a)]. The solution space discussed in Example 1 is a subset of the space defined by the complex medium. Figure 5(b) and (c) shows cross sections of the space where combined (C1+C2+F+H) uptake and oxygen uptake are set at a constant rate of 5 mmol 1 g DCW hr 1, respectively. The line of optimality, P30 in Example 1, is shifted to any of three pathways, P8, P29, and P32, in Example 2. These three pathways obtain similar growth yields but exhibit different behaviors in terms of substrate uptake. With slightly smaller growth yields, extreme pathways P5, P6, P30, and P31 also bound the space and are in close proximity to the optimal pathways. The cross section where the oxygen uptake rate is constant (bottom) is also unbounded, as shown by the dotted lines. Discussion Constraints-based models have been formulated to analyse, interpret, and predict cell behavior from reconstructed metabolic networks under given environmental conditions. They have given useful and surprisingly predictive results (Edwards & Palsson, 2000; Edwards et al., Fig. 4. Aerobicgrowth of the sample system in a complex medium. (a) A table summarizing the 26 extreme pathways which are feasible for this enviroment. The four boxes indicate significant pathway groupings based on by-product secretion and yield. Groups 1 and 2 are the sets of pathways with the highest growth yield. (b) Extreme pathway reduction in the complex medium, similar in format to Fig. 3(a). Note that no environmental constraints are imposed upon the space in this example, and reduction of the solution space is therefore only due to regulatory constraints. Here the pathways have been normalized by the combined uptake of C1, C2, F, and H in order to be represented together in the plots shown in (c) and (d), which are also similar in format to their counterparts in Fig. 3(c) and (d), respectively. In (c), five extreme pathways lie inside the three-dimensional projection (dashed lines). "

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10 318 M. W. COVERT AND B. O. PALSSON

11 REGULATION: SOLUTION SPACE REDUCTION ). The constraints imposed on the reconstructed genome-scale metabolic networks thus far have been the invariant constraints associated with stoichiometry, thermodynamics, and enzyme capacity (Palsson, 2000). These may be considered as first-generation constraints-based models of metabolism. Here we continue to develop second-generation constraints-based models by including the temporary and self-imposed constraints associated with regulation of gene expression and environmental conditions that further constrain allowable functions of the network. A Boolean logicformalism was used to describe regulation and derived logicregulatory rules may be used in conjunction with extreme pathway analysis to examine the effects of regulation on the allowable range of network function. The most important finding of this study is that the imposition of regulatory constraints significantly reduces the size of the solution space. For the skeleton core metabolic network used, the number of extreme pathways was reduced from 80 to as few as two, in some cases, as a result of the imposition of relatively simple environmental and regulatory constraints. In a simulated rich medium, the skeleton network was unconstrained with respect to the environment and yet 67.5% of the pathways were eliminated by regulation. This large reduction in the solution space seems to indicate that despite the complex interaction of many genes to produce an integrated cellular function, simple behaviors can relatively easily be selected by the cell. Another noteworthy observation that can be made from these results regards the extreme pathways that remain after regulatory constraints are applied. In both examples given here, many fermentative pathways were left available to the network despite the presence of oxygen. Although these extreme pathways may be unused, the ability to implement them without the delays associated with transcription and translation would give the cell the ability to rapidly adapt to oxygen deprivation. Finally, the close proximity and even overlap of optimal and near-optimal extreme pathways to one another in the three- and two-dimensional projections of the solution space suggest that the system has numerous means by which to obtain its growth objectives. Extreme pathway analysis of metabolicnetworks at the genome scale has indicated a high degree of pathway redundancy (Papin et al., 2002). Although regulatory constraints greatly reduce such redundancy as has been shown, it seems that a certain amount of flexibility is beneficial to the cell. Such flexibility may be useful to an organism, for example, in colonizing diverse and changing environments. In summary, the application of transcriptional regulatory constraints to metabolic networks results in a large reduction of behaviors available to the network under a given environment. The present study leads to the formulation of a second generation of constraints-based models that can be used to interpret how regulation is used to keep a restricted portion of the total solution space accessible, and thereby, by the process of elimination, force a particular set of phenotypicbehaviors to be expressed. The authors would like to acknowledge Christophe Schilling for helpful discussions and Markus Herrgard for assistance in drawing the figures. This work was funded by the NIH (GM57089) and the NSF (MCB , BES , and BES ). REFERENCES Bailey, J. E. (2001). Complex biology with no parameters. Nature Biotechnol. 19, Bonarius, H.P.J.,Schmid, G.&Tramper, J. (1997). Flux analysis of underdetermined metabolicnetworks: the quest for the missing constraints. Trends Biotechnol. 15, Covert, M. W., Schilling, C. H., Famili, I., Edwards, J. S., Goryanin, I. I., Selkov, E. & Palsson, B. O. (2001a). MetabolicModeling of Microbial Strains in silico. Trends Biochem. Sci. 26, Fig. 5. Diagrammatic representation of solution space reduction. (a) Schematic of the two solution spaces superimposed on one another. The solution space of Example 1 is a subspace of the space of the complex medium. (b) Cross section of the solution space on the oxygen uptake rate growth rate plane where the combined (C1+C2+F+H) uptake is set at 5 mmol 1 g DCW hr 1. (c) Cross section of the solution space on the combined (C1+C2+F+H) uptake rate growth rate plane where the oxygen uptake rate is set at 5 mmol 1 g DCW hr 1.

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(2000). Large-scale prediction of phenotype: concept. Biotechnol. Bioeng. 69, Wong, P. Gladney, S. & Keasling, J. D. (1997). Mathematical model of the lac operon: inducer exclusion, catabolite repression, and diauxic growth on glucose and lactose. Biotechnol. Prog. 13, APPENDIX The flux distribution maps for all 80 of the extreme pathways calculated for the sample network are shown. The active fluxes are shown with thick dark lines, inactive fluxes are denoted by thin light dotted lines and the biomass flux is shown with a dashed line. Note that reactions R 5a and R 5b are isozymes. The 21 extreme pathways which are infeasible under every environmental condition are enclosed in boxes.

13 Fig. A1. REGULATION: SOLUTION SPACE REDUCTION 321

14 322 M. W. COVERT AND B. O. PALSSON Fig. A2.

15 Fig. A3. REGULATION: SOLUTION SPACE REDUCTION 323

16 324 M. W. COVERT AND B. O. PALSSON Fig. A4.

17 Fig. A5. REGULATION: SOLUTION SPACE REDUCTION 325